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We can extend the technique of trigonometric substitution to the hyperbolic functions. Use Theorem 2.20 and the identity cosh2x-sinh2x=1to solve each integral in Exercises 87– 90 with an appropriate hyperbolic substitution x=asinhuorx=acoshu.. (These exercises involve hyperbolic functions.)

1x2+4dx

Short Answer

Expert verified

The value of the integral issinh-1x2+C.

Step by step solution

01

Step 1. Given information.

The given integral is1x2+4dx

02

Step 2. Substitution.

To solve the integral,

Letx=sinh(u)dx=cosh(u)du

sinh2(u)+1=cosh2(u).

Now,

1x2+4=14sinh2(u)+4=12sinh2(u)+1=12cosh2(u)=12cosh(u)

03

Step 3. Value of the integral.

Now, we can write,

1x2+4dx=1du=u=sinh-1x2+C

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Most popular questions from this chapter

Why doesn’t the definite integral231-x2dx make sense? (Hint: Think about domains.)

Complete the square for each quadratic in Exercises 28–33. Then describe the trigonometric substitution that would be appropriate if you were solving an integral that involved that quadratic.

x2-5x+1

Why is it okay to use a triangle without thinking about the unit circle when simplifying expressions that result from a trigonometric substitution withx=asinuor x=atanu? Why do we need to think about the unit circle after trigonometric substitution with x=asecu?

Find three integrals in Exercises 21–70 that we can anti-differentiate immediately after algebraic simplification.

True/False: Determine whether each of the statements that follow is true or false. If a statement is true, explain why. If a statement is false, provide a counterexample.

(a) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(b) True or False: The substitution x = 2 sec u is a suitable choice for solving1x24dx.

(c) True or False: The substitution x = 2 tan u is a suitable choice for solving1x2+4dx.

(d) True or False: The substitution x = 2 sin u is a suitable choice for solvingx2+45/2dx

(e) True or False: Trigonometric substitution is a useful strategy for solving any integral that involves an expression of the form x2a2.

(f) True or False: Trigonometric substitution doesn’t solve an integral; rather, it helps you rewrite integrals as ones that are easier to solve by other methods.

(g) True or False: When using trigonometric substitution with x=asinu, we must consider the cases x>a and x<-a separately.

(h) True or False: When using trigonometric substitution with x=asecu, we must consider the cases x>a and x<-a separately.

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